where \(\sigma _j^x\)  and \(\sigma _j^z\)  are Pauli matrices in \(x\)  and \(z\)  axes, respectively.
We note that the Pauli matrix in \(y\)  axis is not needed [1]}, [2]}.
For the quantum system, each classical vertex variable \(s_j=\pm 1\)  in (REF ) is replaced by \(\sigma _j^z\)  for the \(j\) th quantum spin, which is further realized by a qubit.
The two eigenvalues \(\pm 1\)  of the Pauli matrix \(\sigma _j^z\)  correspond to the eigenstates \( | \!+ 1 \rangle \)  and \( |\!-1\rangle \)  where each further represents the spin up state \(|\!\uparrow \rangle \)  and spin down state \(|\!\downarrow \rangle \) .
In total, there are \(2^b\)  possible quantum configurations made by combining the \(2b\)  eigenstates in the form \(|\!\pm 1\rangle \)  of the Pauli matrices \(\lbrace \sigma _j^z\rbrace ^b_{j=1}\) .
We replace \(s_j\)  in the classical Ising Hamiltonian \({\mathbf {H}}_I^c({\mathbf {s}})\)  by \(\sigma _j^z\)  to obtain the quantum version, that is,
\({\mathbf {H}}^q_I =- \sum _{(i,j) \in {\cal E} ({\cal G})} J_{ij} \sigma _i^z \sigma _j^z - \sum _{j \in {\cal V}({\cal G})} h_j \sigma _j^z,\) 
